PomeCorollary 11.10.6.F8EB48E98C63 For every holomorphic structure in the neighborhood of theorem 11.9.4, there is no positive continuous complex homogeneous function of degree two that is both strictly plurisubharmonic off zero and locally strongly convex in the fibers. If that structure admits a function $H$ as in proposition 11.10.5, there is also no positive continuous complex homogeneous function of degree two that is plurisubharmonic off zero and locally strongly convex in the fibers.
Corollary 11.10.6F8EB48E98C63
Proof. For the first assertion, theorem 11.10.4 would produce a smooth complex norm with plurisubharmonic square, contrary to theorem 11.9.4. For the second, use proposition 11.10.5 and the same obstruction. $\square$
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